STEP 1: find angle $ \beta $

To find angle $ \beta $ use The Law of Sines:

$$ \dfrac{ \sin( \beta )} { b } = \dfrac{ \sin( \gamma )} { c } $$

After substituting $b = 50\, \text{cm}$, $c = 100\, \text{cm}$ and $\gamma = 28^o$ we have:

$$ \dfrac{ \sin( \beta )} { 50\, \text{cm} } = \dfrac{ \sin( 28^o )} { 100\, \text{cm} } $$ $$ \dfrac{ \sin( \beta )} { 50\, \text{cm} } = \dfrac{ 0.4695 } { 100\, \text{cm} } $$ $$ \sin( \beta ) \cdot 100\, \text{cm} = 50\, \text{cm} \cdot 0.4695 $$ $$ \sin( \beta ) \cdot 100\, \text{cm} = 23.4736\, \text{cm} $$ $$ \sin( \beta ) = \dfrac{ 23.4736\, \text{cm} }{ 100\, \text{cm} } $$ $$ \sin( \beta ) = 0.2347 $$ $$ \beta = \arcsin{ \left( 0.2347 \right)} $$ $$ \beta \approx 13.576^o $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \alpha + \beta + \gamma = 180^o $$

After substituting $\beta = 13.576^o$ and $\gamma = 28^o$ we have:

$$ \alpha + 13.576^o + 28^o = 180^o $$ $$ \alpha + 41.576^o = 180^o $$ $$ \alpha = 180^o - 41.576^o $$ $$ \alpha = 138.424^o $$

STEP 3: find side $ a $

To find side $ a $ use Law of Cosines:

$$ a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos( \alpha ) $$

After substituting $b = 50\, \text{cm}$, $c = 100\, \text{cm}$ and $\alpha = 138.424^o$ we have:

$$ a^2 = 50^2 + 100^2 - 2 \cdot 50 \cdot 100 \cdot \cos( 138.424^o ) $$ $$ a^2 = 2500\, \text{cm}^2 + 10000\, \text{cm}^2 - 2 \cdot 50 \cdot 100 \cdot \cos( 138.424^o ) $$ $$ a^2 = 12500\, \text{cm}^2 - 2 \cdot 5000\, \text{cm}^2 \cdot \cos( 138.424^o ) $$ $$ a^2 = 12500\, \text{cm}^2 - 10000\, \text{cm}^2 \cdot -0.7481 $$ $$ a^2 = 12500\, \text{cm}^2 - -7480.7557\, \text{cm}^2 $$ $$ a^2 = 19980.7557\, \text{cm}^2 $$ $$ a = \sqrt{ 19980.7557\, \text{cm}^2 } $$$$ a \approx 141.3533\, \text{cm} $$

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